Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $x = \dfrac{p^2 - p - 90}{-2p^2 - 26p - 72} \div \dfrac{-5p + 25}{p + 4} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{p^2 - p - 90}{-2p^2 - 26p - 72} \times \dfrac{p + 4}{-5p + 25} $ First factor out any common factors. $x = \dfrac{p^2 - p - 90}{-2(p^2 + 13p + 36)} \times \dfrac{p + 4}{-5(p - 5)} $ Then factor the quadratic expressions. $x = \dfrac {(p + 9)(p - 10)} {-2(p + 9)(p + 4)} \times \dfrac {p + 4} {-5(p - 5)} $ Then multiply the two numerators and multiply the two denominators. $x = \dfrac { (p + 9)(p - 10) \times (p + 4)} { -2(p + 9)(p + 4) \times -5(p - 5)} $ $x = \dfrac {(p + 9)(p - 10)(p + 4)} {10(p + 9)(p + 4)(p - 5)} $ Notice that $(p + 9)$ and $(p + 4)$ appear in both the numerator and denominator so we can cancel them. $x = \dfrac {\cancel{(p + 9)}(p - 10)(p + 4)} {10\cancel{(p + 9)}(p + 4)(p - 5)} $ We are dividing by $p + 9$ , so $p + 9 \neq 0$ Therefore, $p \neq -9$ $x = \dfrac {\cancel{(p + 9)}(p - 10)\cancel{(p + 4)}} {10\cancel{(p + 9)}\cancel{(p + 4)}(p - 5)} $ We are dividing by $p + 4$ , so $p + 4 \neq 0$ Therefore, $p \neq -4$ $x = \dfrac {p - 10} {10(p - 5)} $ $ x = \dfrac{p - 10}{10(p - 5)}; p \neq -9; p \neq -4 $